Fix a prime number \(p\) . Let \(K=\mathbb {F}_p(t_0,t_1,\dots )\)  be the field of rational functions over \(\mathbb {F}_p\)  in countably many algebraically independent variables. For each sequence \(\lambda =(\lambda _0,\lambda _1,\dots )\in \mathbb {F}_p^{\infty }\) , set a derivation \(\delta =\delta _\lambda \colon K\rightarrow K\)  by:
\( \delta _\lambda (t_k)=t_{k+1}+\lambda _k t_0\ \ \text{for}\ k=0,1,\dots \) 
This definition induces a well-defined derivation (e.g. see [1]}).
Consider the associated differential polynomial ring \(R=R_\lambda =K[x;\delta ]\) . Observe that \(F=K^p\subseteq {\operatorname{Ker}}(\delta )\) , so \(F=\mathbb {F}_p(t_0^p,t_1^p,\dots )\)  is a central subfield of \(R\) . Obviously \(F\)  is countable. Notice that \(R\)  is \(F\) -affine, and in fact \(R=F\left<t_0,x\right>\) . Indeed, assume by induction that \(t_0,\dots ,t_k\in F\left<t_0,x\right>\)  for some \(k\ge 0\) ; then:
\(t_{k+1}=xt_k-t_kx-\lambda _kt_0\in F\left<t_0,x\right>.\) 
Moreover, observe that \(K\)  is spanned over \(F\)  by the monomials:
\( \lbrace  t_{i_1}^{d_1}\cdots t_{i_r}^{d_r}\ |\ r\ge 1,\ i_1<\cdots <i_r,\ 0\le d_1,\dots ,d_r\le p-1 \rbrace  \) 
as an \(F\) -vector space. Hence \(R=F\left<t_0,x\right>\) . Since \(K\)  is a field, it immediately follows that \(R\)  is a Noetherian ring by [2]}. Hence for each sequence \(\lambda \in \mathbb {F}_p^{\infty }\)  we obtain an affine Noetherian \(F\) -algebra \(R_\lambda \)  as described above.
Fix \(\lambda \in \mathbb {F}_p^{\infty }\) . For any \(f\in R_{\lambda }\)  such that \(f^p=t_k^p\) , write \(f=\sum _{i=0}^{m} c_ix^i\)  with all \(c_i\in K\) . If \(c_m\ne 0\)  and \(m>0\) , it is easy to see that the highest non-zero term in \(f^p\)  is \(c_m^px^{mp}\) , so \(m=0\)  and \(f\)  lies in \(K\) . It is clear that the only \(p\) -th root of \(t_k^p\)  in \(K\)  is \(t_k\)  itself.
Fix \(\lambda ,\lambda ^{\prime } \in \mathbb {F}_p^{\infty }\) . Let \(\phi \colon R_\lambda \xrightarrow{} R_{\lambda ^{\prime }}\)  be an \(F\) -algebra isomorphism; in particular, \(\phi (t_k^p)=t_k^p\) . By the above argument, \(\phi \)  fixes \(K\) . Write \(\phi (x)=\sum _{i=0}^{m} c_ix^i\)  with all \(c_i\in K\)  and \(c_m\ne 0\) . Then:
\( K[x;\delta _{\lambda ^{\prime }}]=K+K\phi (x)+K\phi (x)^2+\cdots \) 
so we can write \(x=\sum _{j=0}^{r} d_j \phi (x)^j\)  with all \(d_j\in K\)  and \(d_r\ne 0\) . The highest non-zero term of the right hand side of the last equation is then \(d_rc_m^rx^{mr}\) , so \(mr=1\)  and \(m=1\) . Now \(\phi (x)=c_0+c_1x\)  where \(c_1\ne 0\) . Then:
\( [\phi (x),t_k]=\phi (t_{k+1}+\lambda _k t_0)=t_{k+1}+\lambda _k t_0 \) 
but:
\([\phi (x),t_k]=[c_1x,t_k]=c_1xt_k-t_kc_1x=c_1\delta _{\lambda ^{\prime }}(t_k)=c_1t_{k+1}+c_1\lambda ^{\prime }_kt_0\) 
Since \(\lambda ^{\prime }_k\in \mathbb {F}_p\) , we always have that \(t_{k+1}+\lambda ^{\prime }_kt_0\ne 0\) , so:
\( c_1 = \frac{t_{k+1}+\lambda _k t_0}{t_{k+1}+\lambda ^{\prime }_k t_0} \) 
Take \(k\gg 1\)  such that \(c_1\in \mathbb {F}_p(t_0,\dots ,t_k)\) , then:
\( 0 = \frac{\partial c_1}{\partial t_{k+1}} = \frac{\partial }{\partial t_{k+1}}\left(\frac{t_{k+1}+\lambda _k t_0}{t_{k+1}+\lambda ^{\prime }_k t_0}\right) = \frac{t_0(\lambda ^{\prime }_k-\lambda _k)}{(t_{k+1}+\lambda ^{\prime }_kt_0)^2} \) 
hence \(\lambda _k=\lambda ^{\prime }_k\) .
Identify two sequences \(\lambda \sim \lambda ^{\prime }\)  if they agree for almost all slots. Since \(|\mathbb {F}_p^{\infty }/\sim ~|=2^{\aleph _0}\) , the claim follows.
